1. Introduction
The onset of precipitation in clouds is the development of particles large enough to sediment relative to cloud droplets and ice crystals. The rate of descent of these precipitation particles is dependent on their mass: larger, heavier particles tend to fall faster than smaller, lighter particles in quiescent conditions. This fall speed difference leads to differential sedimentation of precipitating particles, which explains the frequent appearance of big drops beneath developing ordinary convective clouds preceding the onset of heavier precipitation.
Numerous microphysical processes affect the evolution of the raindrop size distribution (DSD) as the drops sediment, including coalescence, breakup, and evaporation. However, this study will focus only on size sorting of raindrops. Consider a continuously precipitating cloud. If only hydrometeor fallout is considered, differential sedimentation is transient. This is because, after sufficiently long times, the smallest drops have reached the surface, and all drop sizes occupy all altitudes between the cloud and the ground, thereby eliminating any size sorting. However, in nature, various types of atmospheric flows can maintain this otherwise transient size sorting, resulting in prolonged regions of ongoing particle sorting in precipitating storms. For example, consider the presence of a convective updraft. If upward vertical velocities are sufficiently strong, smaller raindrops may be lofted while larger drops are able to fall against the updraft and reach the ground. The presence of the updraft thus maintains the initial transient size sorting by completely removing the smallest particles from the distribution.
Although undetectable to single-polarization radars, size sorting can have an impact on dual-polarization radar observations. The differential reflectivity factor ZDR, first introduced by Seliga and Bringi (1976), is the logarithmic ratio of received signal powers at horizontal and vertical polarizations. It provides bulk information about the power-weighted shape of scatterers in the radar sampling volume. Because raindrop oblateness increases with increasing size (e.g., Pruppacher and Pitter 1971; Brandes et al. 2004a, 2005), ZDR in rain is a reflectivity-weighted measure of raindrop size in the sampling volume. Because DSDs altered by size sorting result in regions of the storm with large median drop sizes, polarimetric observations of these regions reveal large ZDR values. However, in rainfall, ZDR tends to increase with increasing radar reflectivity factor at horizontal polarization ZH (e.g., Sachidananda and Zrnić 1987) because heavier rain tends to have larger concentrations of bigger drops that increase both ZH and ZDR. Thus, ZDR values alone are not sufficient to diagnose regions of ongoing size sorting. Rather, owing to the narrowing effect of size sorting on DSDs (e.g., Rosenfeld and Ulbrich 2003), ZH tends to be relatively low in cases of size sorting. Therefore, observations of relatively low ZH and large ZDR, indicating a DSD skewed to larger drop sizes in relatively low concentrations, may be indicative of ongoing size sorting. The importance of such features led to the inclusion of a “big drops” category in the polarimetric hydrometeor classification algorithm to be implemented in the nationwide Weather Surveillance Radar-1988 Doppler (WSR-88D) radar network dual-polarization upgrade (e.g., Straka et al. 2000; Park et al. 2009). Note that the regions of small drops “sorted out” also represent a skewed DSD that is a result of size sorting, but such small-drop-dominated distributions may not be as apparent in polarimetric radar data.
Size sorting by updrafts and other mechanisms, including strong vertical wind shear, has been recognized for several decades. One of the earliest works to realize the importance of wind shear on precipitation particle size sorting is that of Marshall (1953). He computed analytic trajectories of particles falling in a sheared flow, determining that linear vertical shear results in parabolic particle trajectories, and realized the importance of the particle’s rate of descent for the slope of the trajectory. Further, he stated “as the precipitation pattern moves past a fixed point on the ground, the first precipitation to arrive should be the fastest falling” (Marshall 1953). In the complete absence of vertical wind shear, Marshall demonstrated the absence of any continuous precipitation sorting (aside from the initial transient effect), even for vertically homogeneous winds of any magnitude.
Atlas and Plank (1953) also discuss the differential advection of particles in sheared flow and the importance of gravitational sorting of different sized particles. Gunn and Marshall (1955) computed the effect of wind shear sorting on radar reflectivity and rainfall rate. Their simplified computations produced patterns that bear a remarkable resemblance to those produced by the bin models in the present study shown below. In Gunn and Marshall, the largest raindrops are shown at the leading edge of their modeled rain shaft, foreshadowing the impact of size sorting on ZDR demonstrated herein. Atlas and Chmela (1957) explicitly mention that updrafts can cause additional size sorting of hydrometeors, in addition to wind shear, the latter of which is also mentioned in Hitschfeld (1960). Such conclusions are echoed in later works by Zawadzki and de Agostinho Antonio (1988) and Kollias et al. (2001), which are based on more sophisticated observations and analysis techniques, and that of Battan (1977), which emphasizes the deviation in the DSD from exponential caused by wind shear size sorting.
Sauvageot and Koffi (2000) indirectly hint at size sorting by wind shear as a factor in creating multimodal DSDs in some circumstances. However, their explanation of the appearance of large drops at the leading edge of squall lines invokes overlapping of rain shafts from several convective cells in various stages of growth, but such observations can be explained in much simpler terms by size sorting alone, as shown below. The review by Rosenfeld and Ulbrich (2003) briefly mentions the impact of size sorting on the DSD, as well as the possibility of updrafts affecting the observed DSD at the ground. It is unclear why their schematic indicates a decrease in the number of large drops associated with a substantial increase in the number of medium-sized drops; the narrowing effect of the DSD is exaggerated to the point of perhaps being inaccurate in their presentation.
Scientific curiosity about, and focus on, the issue of differential sedimentation and size sorting has been recently reignited by research in bulk microphysics parameterization schemes (e.g., Wacker and Seifert 2001; Milbrandt and Yau 2005a, hereafter MY05a; Dawson et al. 2010; Mansell 2010; Milbrandt and McTaggart-Cowan 2010, hereafter MM10) and in polarimetric radar observations (e.g., Ryzhkov et al. 2005; Kumjian and Ryzhkov 2008a, 2009, hereafter KR09). The purpose of this study is to review common size-sorting mechanisms and quantify the impact of such sorting on the polarimetric radar variables with the use of simplistic bin models. Additionally, the inability of simple one- and two-moment bulk microphysics parameterizations to reproduce realistic hydrometeor size sorting is discussed, and errors in the computed polarimetric radar variables owing to the assumptions in such bulk schemes are quantified.
2. Size sorting in bulk and bin model configurations
Raindrop terminal fall speed as a function of diameter. Observations of Gunn and Kinzer (1949) are shown as open circle markers. The Atlas et al. (1973) exponential relation (gray solid curve), Atlas and Ulbrich (1977) power-law relation (gray dotted line), Brandes et al. (2002) polynomial relation [Eq. (1); thick black line], and the functional relation used in bulk microphysics parameterization schemes [“Bulk MPS,” thick gray dashed line, from Uplinger (1981)] are overlaid.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
Recent investigations of supercell storms have revealed repetitive polarimetric radar signatures that are seemingly characteristic of such storms (Kumjian and Ryzhkov 2008a, hereafter KR08a). Hypothesized explanations of several of these signatures (notably, the “ZDR arc” and “ZDR column”) invoke size sorting as a significant process contributing to their appearances (KR08b; KR09). Evidence in support of the size-sorting hypothesis has come from the use of Doppler spectra data (Yu et al. 2009), disdrometer observations (Carey et al. 2010), and numerical simulations with bulk microphysics schemes (Jung et al. 2010), as well as simplified bin models (KR09). As shown below, size sorting occurs not just in supercell storms but can appear in any precipitating system. The dilemma is then readily apparent: if size sorting has such a pronounced effect on dual-polarization observations, and bulk microphysics schemes widely used in operational numerical weather prediction models cannot adequately capture size sorting, then there exist serious problems for future attempts to utilize dual-polarization data in storm-scale numerical models (e.g., data assimilation).
The remainder of this paper is devoted to quantifying the impact of size sorting on dual-polarization radar variables using simplified bin models applied to common size sorting mechanisms. Additionally, the errors in the radar variables computed based on assumptions in bulk microphysics schemes will be quantified.
3. Size sorting models
For each subsection below, a model of a particular size sorting mechanism is developed and results are presented. Each mechanism is applied to both a bin model framework and to the assumptions of bulk schemes. The resulting DSDs are converted into S-band polarimetric radar variables as follows. For the bin models, drops are divided into 80 bins (0.05–7.95 mm in 0.1-mm increments). Calculation of the polarimetric radar variables from the bulk scheme profiles uses the same drop partitioning to discretize the moment integrals to ensure that differences in the polarimetric variables are entirely because of differences in the explicit versus parameterized treatment of size sorting. The radar variables are then computed from scattering amplitude calculations using a T-matrix code. Raindrops are assumed to be pure liquid water at a temperature of 20°C, with mean canting angle of 0° with respect to the vertical and a canting angle distribution rms width of 10° (e.g., Ryzhkov 2001; Ryzhkov et al. 2002). The corrected Brandes et al. (2004a,b, 2005) drop shapes are assumed. Such a formulation is analogous to recent polarimetric radar operators developed for bulk models (Jung et al. 2010) and for bin models (Ryzhkov et al. 2011).
a. Pure sedimentation
1) Model description
Differential sedimentation of precipitation particles is the most basic mechanism of size sorting. Following the previously cited literature, a simple one-dimensional model of pure sedimentation is constructed. In this framework, a distribution of raindrops is prescribed at the top of the domain, and these drops begin falling at the initial time. The initial distribution is inverse exponential in shape (i.e., the spectral shape parameter α = 0), with raindrop mass mixing ratio q = 1 g kg−1 and the Marshall–Palmer (Marshall and Palmer 1948) intercept parameter N0 = 8000 m−3 mm−1. Fresh drops are continuously replenished at the top of the domain (“cloud base”) at each time step, which is Δt = 0.5 s. The domain is 3 km tall, with vertical grid spacing of 10 m. There are no changes in air density with height. A simple first-order, forward-in-time, upstream-in-space finite differencing scheme is used. It should be noted that such a scheme is numerically diffusive, thereby smoothing the resulting “shock waves” formed because of the quasi-linear advection equations used in the bulk schemes (e.g., Wacker and Seifert 2001). The fine grid spacing is intended to minimize the effects of the diffusive numerical scheme.
For the bin model, two drop velocity relations are used: the relation in Eq. (1), which is most accurate for larger drop sizes (Thurai and Bringi 2005), and the relation used in computing the moment-weighted fall speeds for the bulk schemes (Table 1, MY05b). The latter is used to make a fair comparison between the bulk and bin model approaches, although it leads to slight underestimation of size sorting because it underestimates the fall speed of the largest raindrops by up to 1 m s−1.
2) Model results
Figure 2a presents the modeled DSDs at t = 333 s at a height of 1000 m above the ground from the bulk scheme simulations as well as the reference bin solutions. This time is selected because it captures the initial transient size sorting effect before the steady-state profiles have developed and the largest raindrops (in the bin framework) have just reached the ground, and the height level is selected because it is near the bottom of the bulk scheme rain shafts. Despite the initial inverse exponential DSD aloft, the resulting bin reference solution DSDs have narrowed, producing a deficit of smaller drops. The 1M scheme (with fixed N0 and α) accounts for decreasing mass at lower levels by removing large drops, doing “violence” to the actual physics [to borrow a phrase from Kessler (1969)]. The 2M scheme (with fixed α) produces a DSD with an extremely shallow slope, also in disagreement with the reference solution. MY05a and MM10 found similar results and proposed diagnostic relations for α, allowing the 2M schemes to capture the narrowing distribution shape owing to size sorting. Alternatively, they demonstrate that 3M schemes (which prognose α) better approximate the narrowing DSD. The 3M scheme solution certainly provides a much better representation of the bin solutions, albeit imperfectly (Fig. 2). The difference in the two fall speed relations used in the bin reference solution is evident in the DSDs at the ground at t = 333 s (Fig. 2b). The Brandes et al. (2002) velocity relation produces higher concentrations of the largest drops compared to the bulk scheme relation (Uplinger 1981; Ferrier 1994; MY05b). There are no DSDs from the 1M and 2M bulk schemes in Fig. 2b because the sedimenting fields of q and NTOT have not reached the ground by this time. In contrast, the 3M scheme produces a narrow DSD similar to the bin solutions.
DSDs from the sedimentation model. The initial DSD aloft is shown by the thin dotted line in both panels. (a) At t = 333 s and height level z = 1000 m AGL, the one-moment bulk microphysics scheme (“1M bulk”) solution is given by the solid black line, the two-moment solution (“2M bulk”) is given by the dashed black line, and the three-moment (“3M bulk”) scheme solution is shown in the solid line with asterisk markers. The reference bin solutions are shown in solid red and blue lines, the difference being the assumed fall speed relation. (b) As in (a), but for surface (z = 0 m) DSDs at t = 333 s from the reference bin solutions. Note that in (a) the bin solutions overlap.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
As expected from the substantial disagreements in the simulated DSDs, the vertical profiles of the S-band polarimetric radar variables also differ significantly (Fig. 3). In the ZH profile (Fig. 3a) neither the 1M nor 2M bulk scheme rain shafts has reached the surface, whereas the surface ZH from the bin solutions is about 36 dBZ. The 3M scheme provides a rather close agreement to the bin solutions, underestimating the ZH by about 2 dBZ at the ground. At midlevels the 2M scheme overpredicts ZH by nearly 10 dBZ, whereas the 1M and 3M schemes match well (<1 dBZ error). Predominantly, this difference is because of the excessive number of large drops (which strongly affect ZH) predicted by the 2M scheme. Note that, where ZH ≤ 0 dBZ, all radar variables have been censored to emulate a minimum detectable radar signal.
Vertical profiles of the S-band polarimetric radar variables predicted by the sedimentation model at t = 333 s. The solutions for the one-moment (solid black line), two-moment (dashed black line), and three-moment (dotted black line) bulk schemes are compared to the reference solutions (red and blue solid curves). Variables shown are calculated for S band: (a) ZH, (b) ZDR, (c) KDP, and (d) ρhv. For the bulk schemes all variables are censored where ZH ≤ 0 dBZ.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
The profiles of ZDR (Fig. 3b) are perhaps the most revealing. The 1M solution predicts an accurate profile down to about 1 km AGL but is entirely wrong below 1 km, owing to a lack of any drops able to fall below that level. Thus, ZH and ZDR decrease sharply at the leading edge of this shock wave. The inability to capture the size sorting is due to the use of a single prognostic moment; all variables of interest are a single-valued function of q and, because q does not reach the ground, ZDR (and all of the radar variables) follows the same pattern. In stark contrast, the 2M scheme produces excessive “size sorting,” resulting in an overprediction of ZDR by over 2.5 dB (a relative error of 168%) at midlevels and toward the bottom of the rain shaft. The simulated ZDR values reach the upper limit (about 4 dB) because of the truncated DSD used to calculate the polarimetric variables (maximum drop size is 8 mm); otherwise, ZDR values could far exceed those observed at S band as unrealistically large (and presumably oblate) drops would be produced. The 3M scheme ZDR profile is closer to the bin solutions, although it still underestimates ZDR by almost 0.5 dB (about 25% relative error). The 3M errors will be discussed in more detail below. Although the reference solutions are similar to one another, the Brandes et al. (2002) velocity relation predicts a surface ZDR value 0.15 dB larger than the power-law relation used in bulk schemes. The increase in ZDR at the surface over the initial value aloft is over 1.0 dB for both bin solutions, illustrating the ability of size sorting to substantially enhance ZDR values in rain.
The profiles of KDP (Fig. 3c) produced by the bulk scheme solutions also show the shock wave problem, albeit smoothed by the diffusive finite differencing scheme used. The 2M scheme produces a midlevel relative maximum in KDP, causing an overprediction compared to the bin solution (relative error of 217%), whereas the 1M scheme’s maximum overprediction of KDP is only about 10%. The 3M scheme overpredicts KDP at midlevels with a maximum relative error of 24%. The reference solution smoothly varies in height and both velocity relations produce nearly identical results. Although differences exist in the profiles of ρhv (Fig. 3d), the changes are small in magnitude at S band (all variations are <0.01) and likely are not measurable. At smaller radar wavelengths, resonance scattering associated with large raindrops (5–6 mm at C band, 3–4 mm at X band) could exacerbate the errors to the extent that they become measurable.
During sedimentation of the sixth-moment Z in the 3M scheme, the shape parameter α can grow to unrealistically large values. Disdrometer observations generally do not reveal α larger than about 15–20 (e.g., Zhang et al. 2001; Cao et al. 2008). The microphysics scheme of MY05b limits α to a maximum of αmax = 40 during sedimentation (D. Dawson 2011, personal communication), which is used in the present study. Varying αmax changes the error characteristics, especially at the bottom of the rain shaft (Fig. 4). Whereas the magnitudes of the relative errors in ZH are <5%, larger errors (>20%) are possible in ZDR and KDP. Above about 1 km AGL, ZDR errors become increasingly negative toward the ground (Fig. 4b), indicating underestimations for all values of αmax. This is because narrowing the DSD by increasing α leads to a decrease in the number of small drops (which is physically consistent with size sorting) as well as a decrease in the number of large drops (which is inconsistent, cf. Fig. 2a). Thus, ZDR is underestimated. Below 1 km, at the very bottom of the rain shaft, the ability of the 3M scheme to reproduce the “true” ZDR profile depends on αmax. Limiting αmax to 10 results in an overprediction of ZDR near the ground because, once the αmax is achieved, the additional size sorting is represented by decreasing the slope parameter Λ, as in the 2M scheme. The vertical profiles of relative errors in KDP (Fig. 4c) demonstrate similar behavior, but of opposite sign. The increasingly positive errors (overestimations) result from an artificial increase in the number of medium-sized drops (cf. Fig. 2a). Based on the excessive narrowing of the DSD that occurs for large αmax demonstrated above, it is recommended to use more rigid constraints (e.g., αmax = 20.0–30.0) for the 3M scheme, especially for polarimetric radar applications.
Relative errors in the polarimetric variables computed from the three-moment scheme with different values of maximum shape parameter α: 10 (black solid line), 20 (dashed black line), 30 (solid gray line), and 40 (dash-dotted gray line). Variables shown are (a) ZH, (b) ZDR, and (c) KDP. The Brandes et al. bin solution is considered “truth” for these error calculations. Positive errors correspond to overestimations by the 3M scheme, negative errors to underestimations.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
b. Vertical wind shear
1) Model description
The effect of vertical wind shear is to provide nonzero storm-relative flow, allowing raindrops to be advected away from directly beneath the cloud. Because smaller drops fall slower than larger drops, the smaller drops encounter this storm-relative flow for longer periods of time and are thus transported farther downstream than the larger drops. This flavor of size sorting, along with that caused by convective updrafts described above, helps explain the enhancement of ZDR (and narrow distribution of large drops) frequently observed at the leading edge of linear mesoscale convective systems (e.g., Ulbrich and Atlas 2007; Morris et al. 2009; KR09; Teshiba et al. 2009).
Schematic of the two-dimensional wind shear model configuration: (left) storm-relative wind (u − ustorm) profile and (right) the domain and example resulting ZH distribution (shaded; dBZ). The “cloud” rainwater mixing ratio q (g kg−1) profile is shown above the domain.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
2) Model results
The results of the bin model configuration are presented in Fig. 6. Left-to-right advection of the precipitation is evident in all fields. Values of all polarimetric radar variables are removed where ZH is less than 0 dBZ, as before. In ZH, the largest values are confined to the center of the rain shaft (Fig. 6a). The ZDR field (Fig. 6b) reveals where the impact of the size sorting is most readily apparent in the polarimetric variables: the highest values are found toward the ground and at the leading edge of the echo along the gradient of ZH. For the shear and initial model parameters used here, the maximal value of ZDR near the surface is 36% larger than the maximal value in the cloud aloft, indicating the ability of wind shear size sorting to amplify the observed ZDR in precipitating systems; also, KDP (Fig. 6c) closely follows the ZH pattern, as expected. The ρhv field (Fig. 6d) exhibits slightly decreased values toward the leading edge, although changes at S band are imperceptible to WSR-88D radars (<0.01).
Results from the two-dimensional wind shear model using the bin formulation: panels show the 2D fields of (a) ZH, (b) ZDR, (c) KDP, and (d) ρhv. Overlaid on (a) are the ZDR contours (0.5–2.5 dB in 0.5-dB increments); (b)–(d) have ZH contours (10–40 dBZ, in 10-dBZ increments) overlaid. For ZH < 0 dBZ, all fields are set to zero.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
The 1M scheme reproduces ZH fairly well, although the rain shaft is narrower at low levels (Fig. 7a). In terms of ZDR (Fig. 7b), the 1M results are wholly unsatisfying owing to the inability of 1M schemes to model size sorting; rather, there is a one-to-one correspondence between q and all other variables (including ZH and ZDR). Therefore, 1M schemes produce the maximum in ZDR collocated with the maximum in ZH. Similar to ZH, the KDP field (Fig. 7c) is reproduced well. Although slight differences are evident in the ρhv field (Fig. 7d), the overall changes in ρhv are too small to be measured at S band.
As in Fig. 6, but for the single-moment bulk scheme configuration.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
Results from the 2M scheme are presented in Fig. 8. Although the width of the rain shaft closer matches the bin model solution, the ZH values (Fig. 8a) are incorrectly enhanced by about 10 dBZ in the middle of the echo below about 2.5 km. The same overprediction of ZH is evident in the pure sedimentation model (cf. Fig. 3a) and is a result of the increased number concentration of larger drops, which significantly contribute to the overall ZH signal (at S band, ZH is proportional to D6). The excessive amount of larger drops is caused by a combination of the decreased intercept parameter N0 (owing to substantially decreased total number concentration NTOT) and a decreased slope parameter Λ (flattening of the distribution shape to account for the still-appreciable rainwater mixing ratio q). These differences are caused by the different sedimentation rates of NTOT and q, owing to the weighted fall speed differences discussed above. Figure 9 depicts the modeled q and NTOT fields, which clearly demonstrate the impact of the different weighted fall speeds on the advection and sedimentation of precipitation modeled in the 2M bulk scheme. The NTOT field, which has a lower fall speed characteristic of the smaller raindrops, is quickly blown downstream, while the q field is advected less rapidly owing to its larger characteristic fall speed (more heavily weighted by the large drops).
As in Figs. 6 and 7, but for the two-moment bulk scheme configuration. Note the changes in the color scales of ZH, ZDR, and KDP.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
Fields of rainwater mixing ratio q (g kg−1), contoured in black, and total number concentration NTOT (m−3), contoured in gray, from the 2D wind shear model run using the two-moment scheme shown in Fig. 8.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
The excessive overprediction of large drops and underprediction of smaller drops yields excessive ZDR over much of the rain shaft (Fig. 8b): ZDR reaches its maximum value at S band. Whereas the 1M scheme produces no size sorting at the leading edge of the precipitation echo, the 2M scheme exaggerates the size sorting (and thus ZDR). Similar to ZH, KDP (Fig. 8c) is enhanced in the center of the echo, overpredicted by 0.6–0.8° km−1. This enhancement can also be attributed to the overprediction of large drop sizes and the strong dependence of KDP on drop size (KDP ~ D4.24). The minimum values of ρhv (Fig. 8d) run along the gradient of ZDR, similar to the bin model results, although again these variations across the precipitation shaft are imperceptible to S-band radars. Note that the overprediction of large drops, which are resonance scatterers at C- and X-band frequencies, will substantially lower the ρhv in regions of high ZDR.
In contrast to the 1M and 2M schemes, the 3M results (Fig. 10) closely resemble the bin model solution. Whereas the ZH field is nearly indistinguishable from the bin model (Fig. 10a), the 3M scheme underestimates the maximum ZDR at the leading edge by nearly 0.5 dB (Fig. 10b). The 2-dB contour of ZDR only reaches a height of about 0.5 km AGL in the 3M scheme, whereas the same contour extends above 1.5 km AGL in the bin solution (cf. Figs. 6a and 10a). Also, KDP values are slightly overestimated in the 3M scheme at low levels (Fig. 10c). As demonstrated in the 1D sedimentation model, the overestimation in KDP occurs because of an overprediction of smaller and medium-sized drops.
As in Figs. 6–8, but for the three-moment bulk scheme configuration.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
The difference fields in ZDR between the bin model solution and those of the bulk schemes are illustrated in Fig. 11. The 1M scheme substantially underpredicts ZDR nearly everywhere (Fig. 11a), especially toward the ground where size sorting is most pronounced in the bin model results. Underpredictions are nearly 2–3 dB through a considerable depth of the rain shaft. The region of slightly negative differences aloft illustrates where the highest q (and thus highest ZH and ZDR in the 1M scheme) is offset from the region of enhanced ZDR in the bin model results. In contrast, large overpredictions of ZDR (>2 dB) are evident in the 2M scheme (Fig. 11b). The narrow region of positive differences at the leading edge is caused by the difference in location of the front edge of the echo. Although the largest drops heavily weight the fall speed of q, the smaller drops still have an impact [Eq. (4)], which causes 
The ZDR difference fields between (a) the bin model and the 1M bulk scheme, (b) the bin model and the 2M bulk scheme, and (c) the bin model and the 3M bulk scheme. Contours in 0.5-dB intervals are overlaid (solid lines for positive differences, dashed lines for negative).
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
As expected, changes in the magnitude of the vertical wind shear affect the maximal ZDR found at low levels in the bin model. Increasing the maximum storm-relative winds further increases the maximum ZDR values near the ground, as well as the vertical depth and width of a given ZDR contour. The relative increase in maximal ground-level ZDR increases approximately logarithmically with vertical wind shear (not shown). Additionally, the DSD aloft controls the relative change in ZDR, as initial DSDs characterized by larger ZDR produce smaller relative changes at the ground (i.e., size sorting acting on DSDs that begin with significant contributions from large drops will produce comparatively smaller low-level ZDR enhancements).
The simulations above consider the case of unidirectional vertical wind shear. It is worth mentioning that size sorting by directional wind shear can produce unique polarimetric radar signatures. Using a simplified three-dimensional bin model of precipitation fallout and advection, KR09 showed that size sorting is capable of explaining the observed “ZDR arc” signature along the forward flank precipitation echo of supercell storms. The strong directional and speed shear common in supercell inflow environments sorts raindrops in such a manner to reproduce the observed shape and alignment of the ZDR arc signature. Further, KR09 found a positive correlation between the simulated magnitude of the maximum ZDR in the signature and the storm-relative environmental helicity in the inflow environment. Thus, unidirectional shear alone can provide enhancements of ZDR at low levels, but the addition of directional shear can also alter the alignment of the observed ZDR enhancement.
4. Polarimetric radar observations
As demonstrated in the preceding simulations, size sorting leads to an increase in ZDR coincident with a decrease in ZH (and KDP). Often, the enhancement of ZDR is located along a gradient of ZH. Indeed, polarimetric radar observations (particularly in convective storms) routinely reveal such patterns of ZH and ZDR that may be attributed to size sorting. For example, Fig. 12 is a genuine range–height indicator (RHI), or vertical cross section, through a mesoscale convective system that demonstrates the impact of size sorting on the observed polarimetric variables at a snapshot in time. Along the leading edge of the squall line, large values of ZDR collocated with modest ZH are found starting at a range of about 65 km through about 80 km. The highest values of ZDR (>4 dB) are found at the base of a developing convective core (evident by the enhanced ZH aloft). The ZDR enhancement owing to size sorting is a result of some combination of transient differential sedimentation and the updraft, the extent of which is unknown owing to the lack of vertical velocity measurements in the storm. The ZDR values of 2–4 dB for ZH < 30 dBZ represent a significant departure (1–3 dB) from what is expected in typical rain DSDs in Oklahoma (e.g., Cao et al. 2008), demonstrating the efficiency of the size sorting process at substantially altering the DSD.
Observations from 0544 UTC 17 Jun 2005 along the azimuth 191°; ZDR (in dB) is shaded, with ZH contours of 30, 40, and 50 dBZ overlaid. The abscissa is range (distance) from the KOUN radar. The high ZDR region near the ground at a range of about 85 km is from biological scatterers (i.e., insects and/or birds).
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
The ZDR arc signature in supercell storms (Fig. 13) is a unique example of size sorting by wind shear, characterized by large ZDR along the gradient in ZH. Indeed, ZDR values in excess of 4 dB are present outside the 30-dBZ ZH contour, indicating a sparse concentration of large drops. The size sorting mechanism hypothesized to produce the ZDR arc signature is the strong wind shear (both directional and speed) in supercell environments (KR08b; KR09). As such, it contains potentially useful information regarding the type of environmental wind shear available to the storm and can serve as an indicator of storm severity (KR09).
Example plan position indicator (PPI) from 0044 UTC 30 May 2004 at 0.5° elevation. The color shading is ZDR (dB), with the 30-, 40-, 50-, and 55-dBZ contours of ZH overlaid. Distances are relative to the location of the KOUN radar.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
Deep convective storms are not the only situations in which size sorting is observed. Figure 14 is a genuine RHI scan from near Bonn, Germany, taken on 22 June 2011 with the Bonn X-band polarimetric radar (BOXPOL), operated by the Meteorological Institute of the University of Bonn. The scan captures an isolated cell producing light to moderate rain falling into an environment with vertical wind shear. The observed fields of ZH and ZDR beneath the melting layer are qualitatively similar to the modeled rain shaft in Fig. 6. Namely, the highest ZDR is located at the leading edge of the shaft along a gradient in ZH (located at about 29 km in range), whereas the higher ZH is offset and coincident with lower ZDR (30–31 km in range). A vertical profile of the Doppler velocities extracted from a range of 29 km (Fig. 15) illustrates the vertical shear, as inbound (negative) velocities become increasingly negative with height. The shear is strongest in the layer near 1 km AGL, where magnitudes reach ~0.02 s−1.
RHI scans of (left) ZH and (right) ZDR taken at 0404 UTC 22 Jun 2011 from the Bonn X-band Polarimetric radar (BOXPOL) operated by the Meteorological Institute at the Universität Bonn (Germany). Data are from the 309.5° azimuth.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
Vertical profile of Doppler velocity extracted from the RHI, shown in Fig. 14, at a range of 29 km.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
Figure 16 depicts a different scenario on the same day when widespread stratiform precipitation was present. At 0944 UTC (Fig. 16, top panels) a distinct melting layer “bright band” is evident between 2.5 and 3.0 km AGL in both ZH and ZDR. At this time, the majority of the rain has not fully reached the ground, evident in the lack of ZH > 20 dBZ below 1 km AGL (strong echoes near the surface are from ground clutter). Enhanced ZDR is observed at the lower portions of the rain curtains, in the gradient of ZH. The next RHI taken 5 min later reveals that all rain has reached the ground (Fig. 16, bottom panels), and the inverse correlation between ZH and ZDR is no longer present. This transient effect of differential sedimentation is reminiscent of the 1D model in section 3a.
As in Fig. 14, but at (top) 0944 UTC and (bottom) 0949 UTC 22 Jun 2011.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
Finally, Fig. 17 is an illustrative example of several cells in various stages of development. The elevated cell located at ~11-km range has the highest ZDR at the bottom of the rain shaft along a gradient of ZH, again indicative of the differential sedimentation flavor of size sorting. The cell at 13-km range again reveals high ZDR at the lower portion of the rain shaft; however, this time the ZH values also increase toward the ground. Rather than indicating ongoing size sorting, the radar observations indicate that the vertical evolution of the DSD in this cell is dominated by some other process, such as raindrop growth by coalescence. Such an example demonstrates that one must assess both ZDR and ZH to detect regions of ongoing size sorting. The example observations from Germany were all collected on the same day, demonstrating that, while size sorting is often transient in nature, it is widespread in many types of precipitating systems.
As in Figs. 14 and 16, but at 1454 UTC 22 Jun 2011.
Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0125.1
5. Discussion
In nature, precipitation formation in continental deep convective storms is generally dominated by ice microphysical processes (e.g., Dye et al. 1974; Rosenfeld and Ulbrich 2003, among others). Thus, melting of graupel and small hail particles often is responsible for the production of large raindrops. A notable exception is in tropical convection where large raindrops forming from efficient coalescence growth in the absence of ice processes are also possible (e.g., Rauber et al. 1991; Szumowski et al. 1997; Hobbs and Rangno 2004). As a consequence of shedding of excess meltwater (Rasmussen and Heymsfield 1987), small hail and graupel of sizes between approximately 8 and 14 mm melt into large raindrops of about 8 mm in size, producing a relative excess of larger drops (Ryzhkov et al. 2009). Such an enhancement of the concentration of the largest drops can amplify the size sorting effect by producing larger ZDR values. This enhancement is probably important for attaining maximal ZDR values because, although ZDR increases monotonically for increasing raindrop size (at S band), the fall speed of drops larger than ~6 mm is essentially the same, according to the Brandes et al. (2002) relation. This leveling off of the fall speeds thereby inhibits further sorting. Additional observations of large drop fall speeds through video disdrometer measurements is critical for better understanding the sorting potential of the largest drop sizes.
As demonstrated above, rain DSDs altered by size sorting tend to be narrow with large median drop sizes (e.g., Battan 1977; Zawadzki and de Agostinho Antonio 1988; Rosenfeld and Ulbrich 2003; MY05a). Narrowing of the DSD is at the expense of the smaller drops, which are sorted out of the distribution. The effect of this process on the DSD is analogous to evaporation (e.g., Rosenfeld and Ulbrich 2003), which also preferentially depletes the smaller drops. Observations of ZDR enhancements are sometimes explained by invoking evaporation as a significant contribution (e.g., Jung et al. 2010). However, Kumjian and Ryzhkov (2010) quantified the impact of evaporation on the polarimetric radar variables in rain and demonstrated that the enhancement of ZDR owing to evaporation (no more than 0.1–0.2 dB at S band, even in extreme cases) is completely dominated by enhancements in ZDR owing to size sorting. Size sorting is more efficient at narrowing the DSD than evaporation.
The results demonstrating the deficiencies of bulk microphysics parameterizations in simulating the polarimetric radar variables in situations of ongoing size sorting have implications for attempts to assimilate polarimetric data. Because of the inability of 1M schemes to produce size sorting, assimilation of ZDR into a model using such a scheme will likely increase analysis errors in cases of size sorting (e.g., Jung et al. 2010). This is because, in the framework of a 1M scheme, there is a one-to-one correspondence between the assimilated observation (ZDR) and the predicted model variable (e.g., rain mass mixing ratio q). Thus, regions of high ZDR would correspond to larger q. In the case of observed size sorting, the model would adjust to the high-ZDR observation by incorrectly increasing rain mass at that location, which is exactly opposite of the physical situation. The incorrect inclusion of additional water mass can affect other processes such as evaporation, which has ramifications for the development and strength of cold pools (e.g., Dawson et al. 2010). Therefore, if using 1M microphysics schemes in assimilation experiments where size sorting may be prevalent (e.g., supercells or other deep moist convection), it is best not to use ZDR data.
The 2M schemes with fixed shape parameter α suffer from excessive size sorting. For this reason, diagnostic-α (e.g., MY05b; MM10) and other techniques (e.g., Mansell 2010) were developed. In MY05a,b and MM10, α increases with increasing mean-mass diameter Dm to reflect the narrowing of DSDs undergoing size sorting. Here ZDR offers an attractive observation that can be related to Dm; of course, high ZDR (and large Dm) alone does not necessarily mean size sorting is occurring. Therefore, it may be desirable to “flag” areas of the storm where size sorting may be occurring using ZH and ZDR observations, thereby limiting the use of such diagnostic-α relations only to regions where they are necessary. Such a flagging system could make use of predetermined ZH and ZDR thresholds, or locations in which the ZH and ZDR data exhibit a strong negative correlation.
Although size sorting may not be widespread throughout all precipitating systems, it can be most pronounced and sustained in deep moist convective storms (DMCS), especially supercells. Many storm-scale data assimilation studies focus on such DMCS, both in observing system simulation experiments (OSSEs) (e.g., Snyder and Zhang 2003; Tong and Xue 2005; Jung et al. 2008; Yussouf and Stensrud 2012) as well as real-data experiments (e.g., Hu et al. 2006a,b; Aksoy et al. 2009; Lim and Sun 2010; Schenkman et al. 2011a,b; Dowell et al. 2011; Snook et al. 2011). Thus, the challenges associated with assimilating polarimetric radar data in cases of vigorous size sorting may be encountered in future endeavors when DMCS are investigated in high-resolution numerical models.
6. Summary
This paper has reviewed size sorting of precipitation particles by the most frequently observed mechanisms, including differential sedimentation, updrafts, and vertical wind shear. Simple bin models were constructed to quantify the impact of size sorting by sedimentation and vertical wind shear on the polarimetric radar variables: ZH, ZDR, KDP, and ρhv. Additionally, the treatment of size sorting by bulk microphysics parameterizations was discussed, and errors in the simulated polarimetric radar variables were quantified.
The following summarizes the key points.
Size sorting of raindrops in the simplified bin models has a significant impact on the polarimetric radar variables, most notably leading to an increase in ZDR along a gradient of ZH. These results are in agreement with previous observational studies (e.g., Ryzhkov et al. 2005; Kumjian and Ryzhkov 2008a; KR08b; KR09).
The initial transient effect of differential sedimentation has been explored thoroughly by the modeling community. However, the transient effect can be maintained by updrafts and vertical wind shear. These size sorting mechanisms have not been investigated widely in the framework of bulk microphysics schemes, but are explored here.
Single-moment parameterizations are incapable simulating size sorting, in agreement with many previous studies (e.g., Wacker and Seifert 2001; MY05a; among others). This inability to reproduce size sorting results in large errors in ZDR computed from the resulting DSDs. Also in agreement with earlier studies, double-moment schemes with fixed shape parameters can suffer from excessive size sorting. This leads to dramatic overestimation of ZDR (by several dB), ZH, and KDP in large parts of a simulated rain shaft encountering wind shear, as well as beneath newly precipitating clouds. Use of a diagnosed shape parameter in a two-moment scheme or a prognosed shape parameter in a three-moment scheme largely mitigates the errors associated with size sorting. However, excessive narrowing of the DSD may occur if the shape parameter is allowed to grow to unrealistic values. Limiting the maximum value of the shape parameter to 20.0–30.0 reduces the errors.
Although size sorting is most apparent in deep convective storms, examples from other precipitation regimes (including stratiform rain and isolated shallow convection) demonstrate that size sorting is widespread in occurrence and is possible in any precipitating system.
Special care must be taken in attempts to assimilate polarimetric radar data into numerical weather prediction models, especially ZDR in cases of ongoing size sorting. Problems may arise because of the fundamental disconnect between the physical process of size sorting (which strongly affects ZDR) and the parameterization of the process in bulk schemes, which in some cases does not resemble reality.
Acknowledgments
Research-grade polarimetric radar data from KOUN are possible because of the tireless efforts of CIMMS/NSSL engineers and scientists who maintained and operated the radar, and for this we are grateful. Partial funding from this work comes from the NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce. The first author would like to thank the members of his doctoral committee for feedback on portions of this work: Drs. Howie “Cb” Bluestein, Guifu Zhang, Alan Shapiro, Eric Abraham (all at OU), and Dušan Zrnić (NSSL). We thank Dr. Edward “Ted” Mansell (NSSL) for reviewing an early draft of the manuscript, as well as Alex Schenkman (OU) for useful discussions and suggestions about data assimilation. Constructive comments and suggestions by Dr. Dan Dawson (NSSL) and two additional anonymous reviewers significantly improved the paper. Data from the BOXPOL radar are courtesy of the Meteorological Institute of the University of Bonn, where the first author held a Doctoral Fellowship funded by the Deutsche Forschungsgemeinschaft (DFG) as part of the Transregional Research Center on “Patterns in Landsurface–Vegetation–Atmosphere Interactions” (TR32).
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